Group #16848552 — subgroups

Subgroups — #16848552

Order 3968.

872 conjugacy classes of subgroups; only the 313 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 313 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
2	#14520952	C₂	1	6	yes	maximal, complemented
3	#14520909	C₂	1	3	yes	maximal, complemented
4	C₄×C₂²⋊Dic₃₁	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₃₁	—	2	8	no	—
7	C₂×#41×C₃₁	C₄	1	8	yes	complemented
8	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₃₁	C₂²	1	3	yes	—
10	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₃₁	C₂²	1	6	yes	—
12	C₂×C₂²⋊Dic₃₁	C₂²	1	6	yes	—
13	Q₈×Dic₃₁	—	2	8	no	—
14	C₂×C₄⋊Dic₃₁	C₂²	1	6	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	6	yes	—
16	C₃₁⋊#95	C₂²	1	1	yes	characteristic
17	C₃₁⋊#95	C₂²	1	3	yes	—
18	C₂⁴×C₃₁	C₂³	1	3	yes	—
19	#41×C₃₁	—	2	32	no	—
20	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
21	C₂×Q₈×C₃₁	D₄	1	8	yes	—
22	C₂×D₄×C₃₁	—	2	12	no	—
23	C₂×D₄×C₃₁	C₂×C₄	1	12	yes	—
24	C₂²×C₄×C₃₁	—	2	12	no	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	12	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
28	C₂²×Dic₃₁	C₂³	1	6	yes	—
29	C₂²×Dic₃₁	C₂³	1	2	yes	—
30	C₂²⋊Dic₃₁	—	2	12	no	—
31	C₂²⋊Dic₃₁	—	4	6	no	—
32	C₄⋊Dic₃₁	—	2	12	no	—
33	C₄×Dic₃₁	—	2	12	no	—
34	C₂³×C₃₁	—	2	6	no	—
35	C₂³×C₃₁	—	2	6	no	—
36	C₂³×C₃₁	—	2	3	no	—
37	C₂³×C₃₁	—	2	3	no	—
38	C₂³×C₃₁	C₂²×C₄	1	6	yes	—
39	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
40	Q₈×C₃₁	C₂²⋊C₄	1	16	yes	complemented
41	D₄×C₃₁	—	2	24	no	—
42	C₂×C₄×C₃₁	—	2	2	no	—
43	C₂×C₄×C₃₁	—	2	24	no	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₃₁	C₂×D₄	1	12	yes	—
47	C₂×Dic₃₁	—	2	2	no	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	2	6	no	—
50	C₂×Dic₃₁	—	2	6	no	—
51	C₂×Dic₃₁	—	2	6	no	—
52	C₂×Dic₃₁	#41	1	4	yes	complemented
53	#2341	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	—	2	6	no	—
55	C₂²×C₃₁	—	2	6	no	—
56	C₂²×C₃₁	—	2	6	no	—
57	C₂²×C₃₁	—	4	6	no	—
58	C₂²×C₃₁	—	4	6	no	—
59	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
61	C₂²×C₃₁	C₂×#41	1	2	yes	—
62	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
64	C₄×C₃₁	—	2	2	no	—
65	C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
66	Dic₃₁	—	2	4	no	—
67	Dic₃₁	—	4	6	no	—
68	C₂²×#41	D₃₁	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	31	2	no	—
70	#390	—	31	6	no	—
71	#380	—	31	3	no	—
72	C₄×C₂²⋊C₄	—	31	3	no	—
73	C₂×C₃₁	—	4	6	no	—
74	C₂×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
76	C₂×C₃₁	#520	1	1	yes	characteristic
77	C₂×C₃₁	#548	1	2	yes	—
78	C₂×C₃₁	#550	1	2	yes	—
79	C₂×#41	—	2	8	no	—
80	C₂×#41	Dic₃₁	1	8	yes	complemented
81	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₃₁	1	3	yes	—
83	C₂³×C₄	C₂×D₃₁	1	3	yes	—
84	C₄×Q₈	—	62	8	no	—
85	C₂×C₄⋊C₄	—	31	6	no	—
86	C₂×C₂²⋊C₄	—	31	6	no	—
87	C₂×C₂²⋊C₄	—	31	6	no	—
88	C₂×C₄²	—	31	6	no	—
89	#95	—	31	1	no	—
90	#95	—	31	3	no	—
91	C₃₁	#2341	1	1	yes	characteristic, complemented, Sylow
92	C₂⁴	C₂²×D₃₁	1	3	yes	—
93	#41	—	2	32	no	—
94	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
95	C₂×Q₈	C₂²⋊D₃₁	1	8	yes	—
96	C₂×D₄	—	2	12	no	—
97	C₂×D₄	C₂×Dic₃₁	1	12	yes	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	—	31	2	no	—
100	C₂²×C₄	—	31	6	no	—
101	C₂²×C₄	C₂×Dic₃₁	1	12	yes	—
102	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
104	C₄⋊C₄	—	62	12	no	—
105	C₂²⋊C₄	—	62	12	no	—
106	C₂²⋊C₄	—	124	6	no	—
107	C₄²	—	62	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	—	2	3	no	—
110	C₂³	—	2	3	no	—
111	C₂³	—	2	6	no	—
112	C₂³	C₂²×Dic₃₁	1	6	yes	—
113	C₂³	C₂³×D₃₁	1	1	yes	characteristic
114	Q₈	C₂²⋊Dic₃₁	1	16	yes	complemented
115	D₄	—	2	24	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	31	4	no	—
119	C₂×C₄	—	62	2	no	—
120	C₂×C₄	—	62	2	no	—
121	C₂×C₄	—	62	6	no	—
122	C₂×C₄	—	62	6	no	—
123	C₂×C₄	—	62	6	no	—
124	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
125	C₂×C₄	C₂²×Dic₃₁	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₃₁	1	12	yes	—
127	C₂²	—	2	6	no	—
128	C₂²	—	2	6	no	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	4	6	no	—
132	C₂²	#41×D₃₁	1	2	yes	—
133	C₂²	#11758566	1	1	yes	characteristic
134	C₂²	#11758567	1	1	yes	characteristic
135	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
136	C₂²	C₂²×C₂²⋊D₃₁	1	2	yes	—
137	C₄	—	2	2	no	—
138	C₄	—	62	4	no	—
139	C₄	—	124	6	no	—
140	C₄	C₂×C₂²⋊Dic₃₁	1	12	yes	—
141	C₂	—	4	6	no	—
142	C₂	C₂²×C₂²⋊Dic₃₁	1	1	yes	characteristic
143	C₂	#41×Dic₃₁	1	1	yes	characteristic
144	C₂	#14521532	1	1	yes	characteristic
145	C₂	#14521706	1	2	yes	—
146	C₂	#14521718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₃₁

C₂

yes

maximal, complemented

#14520952

C₂

yes

maximal, complemented

#14520909

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₃₁

C₂

yes

maximal, complemented

C₂²×#41×C₃₁

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

—

C₂×#41×C₃₁

C₄

yes

complemented

C₂²×Q₈×C₃₁

C₂²

yes

characteristic

C₂²×D₄×C₃₁

C₂²

yes

—

C₂³×C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

Q₈×Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

characteristic

C₃₁⋊#95

C₂²

yes

—

C₂⁴×C₃₁

C₂³

yes

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂×Q₈×C₃₁

D₄

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

C₂²⋊C₄

yes

complemented

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂²×D₄

yes

—

C₂²×C₃₁

C₂×#41

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

characteristic

C₂²×C₃₁

2₋¹⁺⁴

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

—

Dic₃₁

—

Dic₃₁

—

C₂²×#41

D₃₁

yes

characteristic

C₂×C₄×Q₈

—

#390

—

#380

—

C₄×C₂²⋊C₄

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₃₁

C₄×#41

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#550

yes

—

C₂×#41

—

C₂×#41

Dic₃₁

yes

complemented

C₂²×Q₈

C₂×D₃₁

yes

characteristic

C₂²×D₄

C₂×D₃₁

yes

—

C₂³×C₄

C₂×D₃₁

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

C₃₁

#2341

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₃₁

yes

—

#41

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂×Q₈

C₂²⋊D₃₁

yes

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×Dic₃₁

yes

—

102

C₂²×C₄

C₂²×D₃₁

yes

characteristic

103

C₂²×C₄

C₂²×D₃₁

yes

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

124

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

C₂²×Dic₃₁

yes

—

113

C₂³

C₂³×D₃₁

yes

characteristic

114

Q₈

C₂²⋊Dic₃₁

yes

complemented

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×Dic₃₁

yes

—

125

C₂×C₄

C₂²×Dic₃₁

yes

—

126

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#41×D₃₁

yes

—

133

C₂²

#11758566

yes

characteristic

134

C₂²

#11758567

yes

characteristic

135

C₂²

C₂³×Dic₃₁

yes

characteristic

136

C₂²

C₂²×C₂²⋊D₃₁

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

124

—

140

C₄

C₂×C₂²⋊Dic₃₁

yes

—

141

C₂

—

142

C₂

C₂²×C₂²⋊Dic₃₁

yes

characteristic

143

C₂

#41×Dic₃₁

yes

characteristic

144

C₂

#14521532

yes

characteristic

145

C₂

#14521706

yes

—

146

C₂

#14521718

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting, prime radical	C₂²×#41×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂²×#41	D₃₁
hypocenter	C₃₁	#2341
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting, prime radical

C₂²×#41×C₃₁

C₂

socle, cosocle

C₂³×C₃₁

C₂⁴

hypercenter

C₂²×#41

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂²×#41

N	H ≅ G/N	type
1. C₂×Q₈×Dic₃₁	C₂	semidirect
2. #14520952	C₂	semidirect
3. #14520909	C₂	semidirect
4. C₄×C₂²⋊Dic₃₁	C₂	semidirect
7. C₂×#41×C₃₁	C₄	semidirect
40. Q₈×C₃₁	C₂²⋊C₄	semidirect
52. C₂×Dic₃₁	#41	semidirect
80. C₂×#41	Dic₃₁	semidirect
91. C₃₁	#2341	semidirect
114. Q₈	C₂²⋊Dic₃₁	semidirect

H ≅ G/N

type

1. C₂×Q₈×Dic₃₁

C₂

semidirect

2. #14520952

C₂

semidirect

3. #14520909

C₂

semidirect

4. C₄×C₂²⋊Dic₃₁

C₂

semidirect

7. C₂×#41×C₃₁

C₄

semidirect

40. Q₈×C₃₁

C₂²⋊C₄

semidirect

52. C₂×Dic₃₁

#41

semidirect

80. C₂×#41

Dic₃₁

semidirect

91. C₃₁

#2341

semidirect

114. Q₈

C₂²⋊Dic₃₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₈×Dic₃₁	2
#14520952	2
#14520909	2
C₄×C₂²⋊Dic₃₁	2
C₂²×#41×C₃₁	2
#2341	31

subgroup

index

C₂×Q₈×Dic₃₁

#14520952

#14520909

C₄×C₂²⋊Dic₃₁

C₂²×#41×C₃₁

#2341

Subgroups — #16848552

Conjugacy classes of subgroups

Complements

Maximal subgroups